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THE MARKING SYSTEM IN THEORY 
AND PRACTICE 



iE!>urettt0nal PBgrtiologg iMonograytijg 
No. 10 

The Marking System in Tlieory 
and Practice 



I/E. FINKELSTEIN, a. M. (Cornell) 



(Studies from the Cornell Educational Laboratory, No. 14) 




WARWICK & YORK, Inc. 

lalttmur^, 1. ^. A. 

1913 



^ r\ ^ di pT» 



v^ 






Copyright, 1913 

By 

WARWICK & YORK, Inc. 



©CI.A354358 



EDITOE'S PEEFACE. 

When we consider the practically universal use in 
all educational institutions of a system of marks, 
whether numbers or letters, to indicate scholastic 
attainment of the pupils or students in these institu- 
tions, and when we remember how very great stress 
is laid by teachers and pupils alike upon these marks 
as real measures or indicators of attainment, we can 
but be astonished at the blind faith that has been 
felt in the reliability of the marking system. School 
administrators have been using with confidence an 
absolutely uncalibrated instrument. Only within a 
very few years have serious attempts been made to 
scrutinize the theory of marking or to test by statis- 
tical and experimental procedure the degree of pre- 
cision that could be expected in its use. 

What we need to know is: What are the traits, 
qualities or capacities that we are actually trying to 
measure in our marking systems? How are these 
capacities actually distributed in the body of pupils 
or students? What method ought we to follow in 
measuring these capacities? What faults appear in 
the marking systems that we are now using, and 
how can these be avoided or minimized ? 

This monograph (originally prepared as a mas- 
ter's thesis at Cornell University) is a contribution 
directed toward the answering of these very perti- 
nent questions. In it the author reviews the conclu- 

i 



2 MARKING SYSTEM IN THEORY AND PRACTICE 

sions reached by previous investigators, sets forth 
the underlying theories of marking systems, and, 
finally, demonstrates by a painstaking statistical 
analysis of the marks given in his own institution 
what degree of unreliability and what faults of dis- 
tribution inhere in the ordinary percentile system 
that is employed in most schools and colleges. These 
statistical results must not be thought to be peculiar 
to a particular university, or to universities in gen- 
eral. They will be found, upon examination, to be 
the pattern to which the marking system of any edu- 
cational system will tend to conform, and for this 
reason this study has not a local, but a general signi- 
ficance, and the author's conclusions and recommen- 
dations deserve most careful study by all who are 
concerned in educational administration. 
Ithaca, N. Y., April, 1913. G. M. W. 



CONTENTS. 

Chapter I. 

Page 
Introductory 5 

Chapter II. 

Theoretical Considerations 9 

1. Should marks indicate performance or ability 

or accomplishment? 9 

2. What is the theoretical distribution of the qual- 

ities or traits that marks are to indicate?. . . 11 

3. What is the best method of translating the dis- 

tribution into a scale of symbols ? 16 

Chapter III. 

The Distribution op Marks at Cornell University : 

Combined Results for Numerous Courses 21 

1. Marks given in 1902 25 

2. Marks given in 1903 25 

3. Marks given in 1911 26 

4. Combined Curve of Marks in 1902, 1903 and 1911 28 

Chapter IV. 

The Distribution of Marks at Cornell University : 

Results for Individual Courses 37 

1. Variation produced by change of instructors. . 39 

2. Typical distributions of ''high markers'^ 42 

3. Typical distributions of ''low markers" 49 

4. Peculiarities of distribution in other courses ... 60 

5. Marking system of the College of Law 72 

Chapter V. 
Summary and Conclusions 79 



CHAPTER I. 

INTEODUCTORY. 

The idea of making a careful investigation of the 
statistical and psychological problems underlying 
the assignment of grades or marks to students in 
schools and colleges is of relatively recent date. It 
is within the last decade that serious attention has 
been paid to such queries as : What should the mark 
really represent? Should the mark be based upon 
ability or performance, or even upon zeal and en- 
thusiasm! What is the best set of symbols to repre- 
sent ability or achievement! How are the marks 
given by different teachers or different schools 
actually distributed! Is it possible, by exhibition of 
distributions, or by formal instruction in the theory 
of marking, to increase the fairness and reliability 
of marks! Do students tend to secure the same 
standing under different teachers in the same school 
or to maintain their relative standing when proceed- 
ing from class to class or from school to college! 

From the studies of J. M. Cattell (1905), W. S. Hall 
(1906), Max Meyer (1908), W. F. Dearborn (1910), 
A. G. Smith (1911), A. G. Steele (1911), W. T. Fos- 
ter (1911) and others^ who have discussed various 
phases of these problems has come the demonstra- 
tion that few teachers stop to consider what the 



^For exact references see the bibliography, p. 85. 

5 



b MARKING SYSTEM IN THEORY AND PRACTICE 

marking system under which they work really im- 
plies ; that the variability in the marks given for the 
same subject and to the same pupils by different in- 
structors is so great as frequently to work real 
injustice to the students ; and that the marking sys- 
tem in most common use — the percentage system 
with 100 for a maximum and 60 or 70 as a ^^pass 
mark'^ — is in all probability not the best system. 

If these conclusions be granted, it is evident that 
the reliability of the marking system in any institu- 
tion of learning is a matter for investigation. If the 
teachers in the institution are marking unscientifi- 
cally, or if they are using a system which can be 
shown to be inferior in theory and in practice, then 
these facts should be investigated and a remedy 
sought. Nor may anyone seek refuge in the asser- 
tion that the marks of the students are of little real 
importance. The evidence is clear that marks con- 
stitute a very real and a very strong inducement to 
work,^ that they are accepted as real and fairly exact 
measurements of ability or of performance. More- 
over, they not infrequently are determiners of the 
student's career. They constitute the primary basis 
for election to honorary societies, for the award of 
various academic honors, for advancement from 
class to class, for graduation, and may even deter- 
mine in some measure the student's career after 
leaving the institution in which they have been as- 
signed. 

As Meyer (9, p. 661) remarks, "If different grades were siniply 
means of giving some students notoriety above others, the question 
would be immaterial, for a gentleman does not seek notoriety. But the 



^Colvin (2) shows how the marking system can be used as an in- 
centive if it is well organized and rational. 



INTRODUCTION 7 

grade has in more than one sense a cash value, and if there is no uni- 
formity of grading in an institution, this means directly that values 
are stolen from some and undeservedly presented to others. 

"The result is that, among the members of the faculty as v^^ell as 
among the students, men look at each other with suspicion. That this 
attitude is detrimental to the feeling of unity, to the development of 
a college spirit, is clear to even the most superficial observer. What- 
ever contributes to a greater uniformity of grading contributes directly 
towards more peace, a better mutual understanding, a greater commu- 
nity of purposes among all the members of the institution." 

The purpose of the present study is primarily to 
examine the distribution of marks as found in vari- 
ous colleges and classes in Cornell University. But 
consideration is first given to the relative merits of 
different marking systems, and to the theoretical 
considerations which underlie any scientifically or- 
ganized system. 



CHAPTEE II. 

THEOEETICAL CONSIDEEATIONS. 

Three theoretical problems deserve consideration 
before we set forth the data obtained from our in- 
vestigation of the actual distribution of the marks 
of Cornell University. 

The first problem is : Should marks indicate per- 
formance or ability or accomplishment! The second 
problem is : What is the theoretical distribution of 
the quality or traits that the marks are to indicate? 
The third problem is : What is the best method of 
translating the distribution into a scale of marks f 

1. Should marUs indicate performance or ability 
or accomplishment^ 

In certain cases, where the examiner has before 
Mm merely the results of the examinee's efforts to 
answer a given set of questions or to solve a given 
group of problems, as, for instance, in the examina- 
tions submitted in the Civil Service, it is evident 
that marks must be based upon performance, i. e., 
upon work actually done in the examination, without 
regard to native ability or zeal or previous evidence 
of acquaintance with the subject-matter of the exam- 
ination. 

But in actual school or college work, the teacher 
has more to guide him than performance in examina- 
tion alone. He is able, as a rule, to form some idea 



10 MARKING SYSTEM IN THEORY AND PRACTICE 

of the pupil's native ability. He is able, furthermore, 
to take into account evidence afforded in other ways 
than by the performance, of the pupiPs real knowl- 
edge and acquaintance with the subject-matter of 
the course. Thus, he may be convinced that a cer- 
tain pupil fails to do himself justice in his classroom 
and examination performance. He may then decide 
to raise his mark in such a way as to indicate more 
fairly his accomplishment (as distinguished from 
mere performance). 

The issue then takes the form : Shall the pupil be 
marked according to his ability or according to his 
accomplishment 1 

By ability we mean native endowment, intellectual 
capacity. Accomplishment is certainly very largely 
determined by ability, but it is also determined by 
adequacy of previous preparation, and perhaps still 
more by zeal and effort. College teachers will read- 
ily recognize the type of student whose native ability 
is handicapped by poor training in the preparatory 
school, and all teachers recognize the type of student 
whose ability is not reflected in his accomplishment 
for lack of earnest application. 

To most teachers it seems axiomatic that marks 
should indicate accomplishment, and not ability 
alone. If a capable student shirks his work in 
physics, he must suffer the penalty of a low mark. 
If a dull student passes an examination successfully 
by dint of strenuous application, he is entitled to the 
credit of his accomplishment. 

But the opposite position has been defended by 
some writers. Thus, A. G. Smith (11, p. 384) argues 
that *^ College men have greater mental accomplish- 
ment than the average man, but, measured from the 



THEORETICAL CONSIDERATIONS 11 

standpoint of mental ability or capacity, they are 
not a group so. highly specialized as is commonly 
believed. This is especially true in America, where 
the colleges are filled with students drawn from 
every walk of life.'' He asserts that college grades, 
when properly given, should represent ability rather 
than accomplishment. 

2. What is the theoretical distribution of the 
qualities that marhs are to indicate? 

Let us, for the moment, keep both of these posi- 
tions in mind. What, now, should be the distribu- 
tion of marks of a class, first, when the marks indi- 
cate ability; secondly, when the marks indicate ac- 
complishment ! 

Native ability, from all the evidence at our com- 
mand, behaves like any other biological trait. It fol- 
lows that in any population its distribution is that 
known as the curve of error, the probability curve, 
or Gauss's curve. This curve. Fig. A, is a loell- 




FIGI|ltE A 



shaped symmetrical curve, with a mode at the me- 
diaii"point, with deviations of equal magnitude and 
frequency above and below the median, and with a 
progressive diminution in frequency of occurrence 
as the deviation increases in magnitude. 



12 MARKING SYSTEM IN THEORY AND PRACTICE 

The question now arises whether this curve repre- 
sents truly the distribution of ability in high-school 
and in college students (with whom we are more par- 
ticularly concerned). The chief factor which might 
invalidate the application of this curve to a class of 
any rank above the primary grades is the factor of 
selective elimination. Evidently, the idiot, the imbe- 
cile and the moron are eliminated before the high 
school is reached — the idiot at the outset, the imbe- 
cile early in the grades, the moron perhaps in the 
grammar grades or earlier. Undoubtedly, there oc- 
curs at points in the public school system above the 
grammar grades a still further elimination of dull 
pupils from the lower end of the curve of distribu- 
tion. Those who cannot win promotion in the high 
school cannot reach the college. If, then, the lower 
end, at any rate, of the curve of distribution of 
native ability for the entire population be thus cut 
off by the machinery of the public schools, it might 
appear that the distribution of native ability in the 
college would resemble the curve shown in Fig, B, 
which is obviously skewed to the left. 




FIGURE B. 



Not every one, however, is agreed that elimination 
of this kind does take place. It would appear that 
the elimination is not so severe, at any rate, as is 



THEORETICAL CONSIDERATIONS 13 

often supposed. Judd (7, p. 469), for instance, says: 
*^A study of the large high schools of the city of 
Chicago in its relation to the University of Chicago 
shows that students go to college from every level 
of scholarship above the passing mark. ' ' Similarly, 
Meyer (9, p. 666) writes: ^'College teachers usually 
assert that the curve of distribution is not the nor- 
mal curve, but a skewed curve. . . . They are 
usually ready to explain this by referring to the elim- 
ination of poor scholars in the high schools and lower 
schools. I have considerable doubts as to this elim- 
ination. Is the work done in a high school really so 
much like that done in college that there is a large 
previous elimination of poor college students f 

However, the situation, in our opinion, is not quite 
so simple as this. As the pupil passes from kinder- 
garten to university the standard of ability presup- 
posed and exacted for successful work gradually 
rises. So, also, does the ability of the pupil, and this, 
merely from maturity, quite independently of his 
instruction. Along this path of progressive devel- 
opment of ability arrests occur. The feeble-minded 
are instances of such an arrest in the lower stages 
of mental development. Modern psychology teaches 
us that ^^retardations occur continuously on up the 
years of growth to maturity'' (Huey, 6, p. 43). The 
mean ability of a group of children of a given age, 
therefore, advances with increasing age. The distri- 
bution of capacity of a thousand children at their 
tenth year may, therefore, resemble that of the same 
children at their fifth year, save that the entire group 
tends to move forward to a higher stage. 

When the remnant of this same group arrives at 
college, its arrested members have dropped out by 



14 MARKING SYSTEM IN THEORY AND PRACTICE 

elimination, but the poorest of the surviving mem- 
bers represent a minimal ability just sufficient to 
cross the deadline of university entrance. The dis- 
tribution, then, of this group of college freshmen 
might still be that of the normal probability curve, 
save, of course, that the standard of the whole distri- 
bution now differs from that of the same group when 
pupils in the kindergarten, just as the age-norms in 
the Binet-Simon scale advance progressively in diffi- 
culty. If these contentions be accepted, the elimina- 
tion of the academically unfit by the mill of the pub- 
lic school system does not produce a distribution 
like that of Fig. B, but leaves us, after all, the form 
of distribution shown in Fig. A. 

Hence, if college marks should indicate ability, 
Meyer, Smith, Dearborn, Cattell and others would 
undoubtedly be right when they aver that the proba- 
bility curve should represent the type or pattern of 
the normal curve for the distribution of university 
marks, and it would then become a relatively simple 
matter to lay down rules for the guidance of instruc- 
tors and for the standardization of the marking sys- 
tem for any institution, as has been recently done in 
the University of Missouri. 

But now let us return to the second possibility, 
viz., that marks should indicate, not ability, but ac- 
complishment. Is accomplishment distributed like 
ability? We have argued that accomplishment is 
the result of ability, plus previous preparation and 
zeal. When like zeal or effort is exhibited by two 
pupils, the one dull and the other brilliant, it seems 
probable that the gifted pupil reaps a proportion- 
ately larger result. It is as if effort were multiplied 
into ability rather than added to it. It is difficult to 



THEORETICAL CONSIDERATIONS 15 

prove this assertion, but if it be admitted, the con- 
clusion is evident that, if all the students put forth 
the same amount of effort, the curve of accomplish- 
ment would be skewed to the right {Fig. C). This 




FIGURE C. 

skew would appear in the absence of any marking 
system. But when a marking system is arranged to 
measure accomplishment, another factor undoubt- 
edly enters to skew the curve further to the right. 
This factor is the incentive offered to certain groups 
of students by the critical points of the system. 
Thus, at Cornell University, where a mark of 60 or 
above is necessary to pass, there evidently exists a 
type of student of inferior or average ability who 
aims for this mark. The possibility of exemption 
from final examination by attaining a mark of 85 in 
preliminary examinations offers also a powerful in- 
centive to students of average ability to push their 
accomplishment to the extent of their ability. Again, 
an average student, by persistent effort, may hope to 
obtain an average of 80 or over, and thus be eligible 
for consideration for Phi Beta Kappa and other 
special honors. On the other hand, the brilliant stu- 
dent, who probably profits most by effort, cannot at- 



16 MARKING SYSTEM IN THEORY AND PRACTICE 

tain a mark above 100, and it is not difficult for him to 
obtain a mark of 85 or over. The net result, then, of 
the imposition of a marking system is to crowd the 
grades of accomplishment forward to the right, and 
thus, again, to tend to form a skewed, and not a bell- 
shaped curve. 

Finally, the actual distribution of over 20,000 
marks in Cornell University is a curve skewed to the 
right, as will be shown below (p. 28). If such a 
curve, compounded of many classes and many ex- 
aminers, may be thought to represent with great 
reliability the consensus gentium of the faculty of 
the University with regard to the distribution of 
accomplishment, it affords us one more reason for 
believing that the distribution of accomplishment 
is skewed to the right. The pattern or theoretically 
ideal curve of high-school and college marks is, 
therefore, not the probability curve, but the skewed 
curve with the mode to the right of the middle of the 
abscissa. 

3. What is the best method of translating the dis- 
tribution into a scale of symbols? 

Given now a pattern or an ideal distribution of ac- 
complishment, the further question remains: What 
is the best method of dividing this distribution into 
groups for translating accomplishment into a symbol ' 
or mark ! Theoretically, there are numerous ways of 
making such a division, of translating standing into 
a scale of marks. In actuality serious consideration 
has been given to a few forms of scale only, viz., the 
division into two, into three, into four, into five and 
into one hundred groups. 

(a) The two-division system. Simplest of all 
devices is that which divides all students into two 



THEORETICAL CONSIDERATIONS 17 

groups — ^ ^passed'' and ^'not passed.'' Such a divis- 
ion is in operation at Cornell University, and at 
many other institutions in the reporting of the work 
of students in the Graduate School. While more defi- 
nite marks may be assigned at the discretion of the 
instructor, it is customary to report the work of 
the students simply as satisfactory or unsatisfac- 
tory. There are some members of the faculty who 
would be glad to see this system extended to under- 
graduates, but the majority of college teachers, and 
of students as well, prefer to use a more precise 
scale, despite the greater labor entailed in the grad- 
ing of work and in the keeping of records. It is felt 
that the student should receive a more exact notion 
of his accomplislunent, and that for many extrane- 
ous purposes — selection of members of advanced 
classes, distribution of various awards, etc. — a finer 
scale is necessary. 

(b) The three-division system. E. B. Sargent 
(10, p. 64), who has advocated the placing of a pat- 
tern curve of distribution on all record sheets to 
guide the marking of examiners, argues for a three- 
division system in which the groups are labeled : in- 
ferior, mediocre and superior. The fundamental 
merit of this system is the psychologically correct 
distinction of a large group of students of aver- 
age accomplishment, midway between two smaller 
groups whose work falls short of, or exceeds, this 
average accomplishment. The defects of this sys- 
tem lie, first, in the difficulty of distinguishing fail- 
ure from mere inferiority which is still entitled to a 
pass, and, secondly, as A. G. Smith (11, p. 390) has 
pointed out, ^^that it furnishes no distinguishing 
mark of excellence unless the middle group is made 



18 MARKING SYSTEM IN THEORY AND PRACTICE 

so large as to be open to all the criticism that can be 
urged against a ^pass' and ^not pass' method of 
grading. ' ' 

(c) The four-division system. No one has very 
seriously defended a four-division system. There is 
no psychological nor statistical justification for it. 

(d) The five-division sy stern. The best possible 
division of the marking scale for any small number 
of groups is the five-member division. This plan is 
based upon the orientation of all students around a 
central group whose accomplishment is construed to 
be average or medium. The theory of the distribu- 
tion of ability, and hence in large measure of accom- 
plisliment, teaches us that mediocrity is the com- 
monest condition. The largest single homogeneous 
type of student is the average student. Above and 
below the average lie groups of smaller size contain- 
ing superior and inferior students — superior and in- 
ferior with reference to the average group. The 
five-division system improves upon the three-divis- 
ion system in that it further differentiates the out- 
lying groups ; those superior to the average are sub- 
divided into two groups, the superior and the excel- 
lent or exceptionally good; those inferior to the 
average are subdivided into inferiors and failures.^ 

The' theory of the application of the five-division 
system to the actual grading of students assumes 
that the actual distribution of marks should conform 



^For purposes of administration it may be thought desirable further 
to differentiate between 'conditioned' and, 'absolutely failed' — following 
the custom of many institutions of permitting the former to try for a 
'pass' by taking a 'make-up' examination, but compelling the latter to 
take the course again in its entirety in order to secure credit. In prac- 
tice the scale would then contain six symbols, but it would, neverthe- 
less, be in theory a five-division system. 



THEORETICAL CONSIDERATIONS 19 

fairly closely to a theoretically predetermined dis- 
tribution. The question as to what this predeter- 
mined distribution should be in the case of a skewed 
curve will be treated later (pp. 30-33). We may set 
forth here, however, the plans proposed by certain 
psychologists for the translation into marks of the 
bell-shaped curve of distribution. If we assign the 
symbols A, B, C, D and E to excellent, superior, 
average, inferior and failure, respectively, the divi- 
sions recommended by Professors Meyer, Dearborn 
and Cattell for each hundred students are as follows : 

A. B. C. D. E. 

Meyer 3 22 50 22 3 

Dearborn ' 2 23 50 23 2 

Cattell 10 20 40 20 10 

In the opinion of these writers, then, from 40 to 
50 per cent, should be marked average, from 20 to 
23 per cent, should be superior, and inferior to the 
average, and from 2 to 10 per cent, should receive 
the highest mark and the like number should fail. 
, (e) The percentile system. It is not very diffi- 
cult to grade students on the five-division system. 
Is it possible to say as much of the system which 
many institutions follow, according to which marks 
are based on a scale of 100 points! Theoretically, 
this scale implies that distinctions of a fineness of 
one-hundredth may be made, and in practice such 
distinctions are constantly attempted. But what is 
the difference, if any, between a mark of 75 and one 
of 76? What, for that matter, does 75 mean! Has 
the student accomplished 75 per cent, of some ideal 
accomplishment! It is a commonplace of statistics 
that a scale whose units are not defined or whose 
units are not identical throughout is no scale at all. 



20 MARKING SYSTEM IN THEORY AND PRACTICE 

The fact that different instructors place a different 
interpretation upon the symbols of the percentile 
system is evidence enough that it is not the scientific 
measuring rod that it pretends to be. The very fact 
that its divisions are so minute is doubly insidious ; 
it promises precision, but it cannot afford it/ In 
short, the 100-division scale has no psychological 
justification. On the other hand, the five-division 
system, which is evidently based on a different plan 
altogether, is simple to use, and the results of each 
instructor are easily checked at any time.^ 



^Since this was written, Professor Starch of the University of Wis- 
consin has reported the results of a study upon the "Reliability and 
Distribution of Grades" {Psychol Bulletin, 10, Feb. 15, 1913, p. 74), 
which shows that the marks assigned by more than one hundred teach- 
ers to two papers in English and one in geometry have a probable error 
of from 4.0 to 7.5 points on a percentile scale. Starch concludes : 
"The steps on a scale should be at least twice the size of the mean 
variation or probable error of the measurements in order to be dis- 
tinguishable steps. Hence the steps on a marking scale should be at 
least two times 4.2, or approximately 8 points. And hence on a scale 
of passing grades of 70 to 100 only four steps can be used with any 
degree of objective reliability." This statistical conclusion, then, con- 
firms very prettily our argument for a five-division scale — four marks 
al)ove the passing limit and one mark below it for failures. 

^See, for example, the very interesting history of the establishment 
and operation of a five-division system at the University of Missouri, 
as narrated by Meyer (8, 9). 



CHAPTER III. 

THE DISTEIBUTION- OF MAKKS AT COENELL UXIVEESITY : 
COMBINED EESULTS EOE NUMEEOUS COUESES. 

The purpose of this chapter is to show the actual 
distribution of marks at Cornell University when a 
large number of different courses are combined. 
The effect of combining the marks of several thou- 
sand students in numerous courses is, naturally, to 
eliminate or to cancel chance variable errors in the 
marking. The resultant curve thus compounded of 
the marks given in varied classes is as true a picture 
as can be obtained of the actual distribution of ac- 
complishment of students as judged by the instruct- 
ing staff. Whether the distribution may not be 
affected by certain constant errors is a matter to be 
discussed a little later. 

The material at our disposal for securing these 
combined results comprises three sets of data. The 
first set represents the marks given in the College 
of Arts and Sciences during the first term of 1902- 
03 to 5396 students in 66 courses. The subjects 
represented are Latin, German, French, English, 
Philosophy, Psychology, Education, History and 
Political Science, Mathematics, Physics, Chemistry, 
Botany, Invertebrate Zoology, Physiology and Geol- 
ogy. The second set of data represents the marks 
given in the same College in the following academic 

21 



22 MARKING SYSTEM IN THEORY AND PRACTICE 



ir 



year. It includes the same courses and 7522 stu- 
dents/ The third set of data represents the marks 
collected by the writer and shown in detail in Chap- 
ter IV. It includes 7430 marks in 31 courses in dif- 
ferent colleges of the University (Arts and Sciences, 
Agriculture, Mechanical Engineering and Civil En- 
gineering), and also 711 marks in three courses in 
the College of Law. These three sets of data will be 
referred to for convenience as the 1902 marks, the 
1903 marks and the 1911 marks.^ We shall present 
the distribution for these three sets of data in the 
order given, then combine them into a single distri- 
bution and discuss the form of this final compounded 
curve. 

TABLE I. 

Collective Distributions. 
Showing the Per Cent, of Students in the Several Groups. 

No. of 
Year. Marks. 0-39. 140-44. 45-49. 50-54. 55-59. 60-64. 

1902 5,396 1.6 1.7 1.6 3.4 2.3 11.8 

1903 7,522 1.3 1.2 2.1 3.5 1.8 10.2 

1911 7,430 0.6 0.4 1.1 2.7 2.2 12.8 

All three 20,348 1.2 1.1 1.6 3.2 2.1 11.6 

All three. (Revised).... 20,348 1.2 1.1 1.6 3.2 5.6 8.1 

Ex- 
Year. 65-69. 70-74. 75-79. 80-84. \ 85-89. 90-95.. 95-100. empt. 

1902 10.2 12.9 15.8 15.6 ' 11.5 8.9 2.7 23.1 

1903 10.0 13.6 16.0 16.2 11.6 8.7 3.8 24.1 

1911 12.5 15.4 16.4 12.7 14.0 7.2 2.0 23.2 

All three 10.9 13.9 16.1 14.8 12.4 8.3 2.8 23.5 

All three. (Revised) 10.9 13.9 16.1 14.8 12.4 8.3 2.8 23.5 

^My thanks are due to Prof. W. F. Willcox of Cornell University for 
the use of these two sets of data which were compiled under his direc- 
tion several years ago. 

^As explained in Chapter IV, this third set of data contains marks 
extending backward from June, 1911, for a length of time necessary 
to secure at least 200 marks for each course. In most cases the period 
represented falls between 1910 and 1911. 



RESULTS FOR NUMEROUS COURSES 23 

To make the charts intelligible a word of explana- 
tion is needed. The percentile system is used at 
Cornell — save in the College of Law, of which we 
shall speak more definitely below. The ^pass mark' 
is placed at 60. A mark between 40 and 59, inclusive, 
is known as a ^condition' — the student is entitled 
within one year to try a 'make-up' examination in 
order to gain credit for the course, provided he 
reaches 60 or above in this examination.^ A mark 
below 40 represents complete failure; the student 
must take the course again in its entirety, and must 
then pass at 60 or over in order to gain credit for 
his work. In certain courses students may, at the 
option of the instructor, be exempted from the final 
written examination. The mark which must be at- 
tained to secure exemption varies in different col- 
leges, save that in the College of Arts and Sciences 
exemption, if given at all, must be based upon a 
mark of 85 or over. In the College of Mechanical 
Engineering there are no special or formal final ex- 
aminations ; the mark is based upon the work of the 
students during the term. 

All the charts of distribution are plotted with the 
scale of marks as the abscissas, and in units of 5 (or 
6) points each, with the highest marks at the right. 
Thus, the division at the extreme right represents 
the six marks 95, 96, 97, 98, 99 and 100. The next 
division represents the five marks from 90 to 94, in- 
clusive, and all subsequent divisions on toward the 
left end of the abscissa each represent 1nYe marks, 
save that all marks including 39 and below are in 

^Through oversight, it wos not noticed until too late to make the 
correction that the mark of 40 is counted as complete failure, not as 
conditioned. This error does not affect the conclusions in any way — 
Editor. 



24 MARKING SYSTEM IN THEORY AND PRACTICE 

the single division at the extreme left. The ordi- 
nates in all charts represent the per cent, of students 
receiving the marks in the various divisions of the 
scale. 

In all the charts, furthermore, two additional fea- 
tures of the distribution are shown. First, the per- 
centage of students who would be exempt from the 
final examination, on a basis of 85 or over (assum- 
ing exemption were permitted in all classes), is 
shown in the accompanying legends. Secondly, the 
range of marks received by the middle 50 per cent, 
of the students is shown on each graph by the size 
and position of the solid black area. This range is 
found by counting off 25 per cent, of the marks from 
the upper, and 25 per cent, of the marks from the 
lower end of the total distribution found in each 
case.^ 

Since, by theory, the middle 50 per cent, of any 
group of students must be neither brilliant nor dull, 
but simply straightforward, average students, the 
distribution of this group is of special interest. 
Speaking generally, we should not expect the marks 
obtained by this group to spread over a large range, 
since they represent a homogeneous group in point 

^To avoid possible misunderstanding, a word should be said concern- 
ing the boundaries of the black area on the charts. The upper and 
lower limits of the middle half of the students need not necessarily, 
of course, coincide with a division point upon the abscissa. Thus, in 
Chart I, the upper limit of the middle group lies in the range 80-84, but 
there are also some students in the upper 25 per cent, whose marks fall 
within the same range. To be specific, since 23.1 per cent, lie in the 
range 85-100, 1.9 per cent, of the students superior to the average lie 
in the range 80-84 per cent. Accordingly, the vertical column erected 
as an ordinate over the range 80-84 is blackened to within 1.9 units 
of its tip only, and not fully to the tip. Similarly, a mark lying be- 
tween 65 and 69 is for the most part obtained by students belonging to 
the middle or average group, but there are a few students inferior to 
the average (on our theoretical definition of average) that also obtain 
marks within this range. 



RESULTS FOR NUMEROUS COURSES 



25 



of ability and accomplishment. Again, we should 
not expect an average student to attain exemption 
from final examination, nor, on the other hand, to 
run any grave risk of being ^ conditioned. ' 



zo- 



15 — 



10 — 



5 — 



ZO- 



15- 



10- 



5— 



0- 



CCLLEOE OF AktS AND SciE-NCES IN WOZ' 

66 Courses 5396 Students 
JO. 6 Below Pass ^3.^ ^Exempt 




O 40 45 50 55 GO 65 70 75 60 85 30 95 100 



College of Arts and Sciences in 1903 
66 Courses 75ZZ Students 

9.9^ Below T^ss Z4.I X Exempt 




40 45 50 55 60 65 70 75 80 65 90 95 lOO 



CHARTS I AND 2. Combined Distribution of MARK^s in 
66 Courses in 1902 and 1903. 



"The University in I9lf. 
31 Courses 7430 Stude-nts. 
7^ BelowT^ss Z1>.Z% Exempt. 



15. 



\0 — 



- 




O 40 45 50 55 60 65 70 75 SO 85 90 95 /OO 



i 



CHART 3. Distribution of Marks in 31 Courses 
IN 1911. 



28 



RESULTS FOR NUMEROUS COURSES 27 

The distribution of the three sets of data used for 
combined curves is shown in Table I, in tabular 
form, and in Charts 1, 2 and 3, in graphic form. 
Chart 4 is the combination of Charts 1, 2 and 3. 

It will be noted that the distribution in the three 
sets of data is closely similar. They all take the 
form of a curve skewed to the right, and with an 
evident disturbance in the region of 60, the ^pass' 
mark. 

The combination shown in Chart 4 is worthy of 
special mention. It includes in all 20,348 marks, 
enough surely to entitle it to the designation as the 
true curve^ of the distribution of accomplishment, as 
indicated by the marks of Cornell University- The 
form of the curve is again the skew to the right, not 
the probability curve which, according to Meyer, 
Cattell, Dearborn and others, is the theoretically 
correct distribution. 

In Chart 4, just as in all the three curves from 
which it is compounded, there appears a disturbance 
at 60. There are, apparently, too few marks between 
55 and 59, too many marks between 60 and 64. Two 
factors may contribute to this result. First, as sug- 
gested already, it is probable that a certain number 
of students make a special effort to just secure a 
pass mark. Secondly, many instructors are loath to 



^A. G. Steele (12, p. 525), from an experiment conducted during a 
summer session at Miami University, concludes that "tlie average 
judgment of several competent judges gives approximately the true 
grade, and the one which will be more nearly rightly interpreted by the 
greatest number of judges." 

Similarly, in the experimental study of the psychology of testimony, 
it has been found that the combined result of the evidence of several 
witnesses affords a fairly reliable indication of the true state of affairs. 



zo- 



15- 



/o— 



5- 



0- 



The Combined Distribution. FinalCurve. 
9.^%Beud»vPass ^3.5^ Exempt ^0,348 /V\arKS 
^Dotted Lines Show Revised 
Percents^ 



n 



40 45 50 55 GO 65 70 75 80 85 30 35 >00 



CHART 4. Combined Distribution of the 20,348 Maeks 
Repbesented in Charts 1, 2 and 3. 



28 



RESULTS FOR NUMEROUS COURSES 29 

condition a student by a narrow margin. If the 
work of the term justifies a mark between 55 and 59, 
the instructor frequently raises the mark, arbitrar- 
ily, to 60. He may feel a measure of insecurity in his 
judgment, and he may dislike to suggest to the stu- 
dent the possibility of any argument about the ^ con- 
dition ^ which a mark in the region of 58 would entail. 
The second of these two factors is undoubtedly the 
more potent in distorting the distribution in the 
region of 60.^ The disturbance in the 60 region has 
been eliminated by the method of differences and the 
resulting correction is shown by the dotted lines of 
Chart 4. 

In Chart 1 there is 2.3 per cent, in the 55-59 group, 
and 11.8 per cent, in the 60-64 group. Let 11.8 be 
equal x. Then the 55-59 group is equal to 14.1-^. 
Taking two figures on each side of these numbers, we 
find X in the fifth difference. 

1.6 3.4 14.1— a? (B 10.2 12.9 

1.8 10.7— a? 2a?— 14.1 10.2— a? 2.7 

8.9— a? 3a?— 24.8 24.3— Sa? a?— 7.5 

4a?— 33.7 49.1— 6a? 4a?— 31.8 

82.8— 10a? 10a?— 80.9 

20a?— 163.7 

X is, therefore, equal to 8.1. 
The 60-64 group should then be 8.1, and the 55-59 
group should be 6.0. In the same way we smooth the 
graphs of Charts 2 and 3. For Chart 2 we find that 
the 60-64 group should be 7.1 instead of 10.2, and 
the 55-59 group 4.9 instead of 1.8. For Chart 3 the 
60-64 group is changed from 12.8 to 9.2, and the 



^There are some instructors who deliberately raise the mark of the 
doubtful cases, not merely to 60, but to 65 or even to 70, with the idea 
that if a student is to be passed, he "might as well be passed hand- 
jsonaely," as one of them puts it. 



30 MARKING SYSTEM IN THEORY AND PRACTICE 

55-59 group is changed from 2.2 to 5.8. In Table I 
the data of Chart 4 appear in tabular form both with 
the original and the revised per cents. 

A word should now be said about the average 
mark and the translation of the curve of Chart 4 into 
a five-division marking system distributed around 
the average group. Here we recur to the problem 
broached in Chapter 11 (p. 19), but now with the 
advantage of an actual curve of distribution as the 
basis of discussion. 

The weighted arithmetical mean for these several 
compounded curves is as follows : 1902 marks, 73.0 ; 
1903 marks, 74.12, and 1911 marks, 75.23. We may 
be justified, then, in taking 75 as the net average 
mark of the students here represented for the pur- 
poses of fitting the curve of actual distribution to 
the curve of theoretical distribution. Now, if we as- 
sumed, like Cattell, Meyer and others, that the sym- 
metrical probability curve were the correct theoreti- 
cal distribution, then, if 75 were the median or mode, 
there ought to be equal divisions above and below 
this mode. But the deviations above 75 cannot ex- 
ceed 25 points; hence the lowest possible mark, or 
complete failure, would have to be indicated by the 
mark of 50. 

It is perfectly evident that this conclusion leads 
to an absurdity. Complete ignorance of a subject 
should be represented by a mark of 0, according to 
the implication of the percentile scale. It is true 
that few marks are given below 50; nevertheless, 
such marks are given, and, furthermore, a distinc- 
tion is made, administratively, between marks below 
40 and marks between 40 and 59.^ If a translation of 
the normal probability curve to the percentile scale 

*See note, p. 23. 



RESULTS FOR NUMEROUS COURSES 31 

is to be made, then the meaning of the percentile 
scale must apparently be radically altered. 

If, however, we assume that the correct theoretical 
distribution, for the reasons advanced in Chapter II, 
is a curve skewed to the right, then the translation 
of this curve to the actual distribution shown in 
Chart 4 is feasible. Let us start with the criterion 
for exemption, the mark of 85. On all accounts, an 
accomplishment sufficient to justify exemption from 
final examinations should be of a grade superior to 
the average accomplishment. Exemption should be 
a privilege reserved for superior work. From Chart 
4 we find that 23.5 per cent, of the students of the 
University reach this degree of accomplishment. 
This number corresponds, then, very closely to the 
number that Dearborn and Meyer assign as the num- 
ber that are theoretically superior to the average 
group. 

Secondly, our distribution suggests a differentia- 
tion of the superior group at the mark of 95. A mark 
of 95 or over is gained by 2.8 per cent. ; a mark of 85 
to 94, inclusive, is gained by 20.7 per cent, of the Cor- 
nell students. This division into excellent and supe- 
rior students also corresponds very closely to the 
distribution assigned from theoretical grounds by 
the writers just mentioned. 

Thirdly, in the system used at Cornell the mark 
of 60 serves as another crucial point, A mark below 
60, that is, a ^condition' or a ^failure,' is given in 
actual practice to 9.2 per cent, of the students. Here 
we are unable to arrange a distribution that corre- 
sponds to the theoretical requirements of the proba- 
bility curve according to Meyer and Dearborn, but 
the distribution does closely correspond to Cattell's 



32 MARKING SYSTEM IN THEORY AND PRACTICE 

assignment of 10 per cent, to the gronp of poorest 
students. But we have already seen that the marks 
between 55 and 59 are affected by the operation of 
a constant error. Our correction of this error (see 
p. 29 and Chart 4) brings the number properly as- 
signable to this lowest group to 12.7 per cent. Of 
these marks a subdivision may be made which will 
give 1.2 per cent, to ^complete failures' and 11.5 per 
cent, to * conditions. ' In fine, then, it does not appear 
that the poorest group should be equal in size to the 
best group ; rather that the number of students con- 
ditioned and failed should properly exceed the num- 
ber that receive the mark of highest excellence. 

Fourthly, we have left a group of 63.8 per cent, of 
inferior and average students whose work entitles 
them to pass, but does not entitle them to exemption 
from final examination. The problem is to deter- 
mine the dividing line between inferior and average 
accomplishment. Theoretical considerations compel 
us to assign more students to the average than to 
the inferior group. Inspection of Table I and Chart 
4 shows that a division at 70 yields satisfactory re- 
sults. The inferior group, now receiving marks be- 
tween 60 and 69, inclusive, comprises 19.0 per cent, 
of the students, and is thus practically equal with the 
superior group previously set aside. The average 
group, now receiving marks between 70 and 84, con- 
tains 44.8 per cent, of the students. As noted, Meyer 
and Dearborn would place 50 per cent, and Cattell 
40 per cent, in this group. 

To eliminate the decimals, we may lay down as 



RESULTS FOR NUMEROUS COURSES 33 

the pattern or ideal distribution of marks the fol- 
lowing schema : 

Group. Poorest. Inferior. Average. Superior. Excellent. 

Percent 12 19 45 21 3 

Range in 1 

percentile )■.... 0-59 60-69 70-84 85-94 95-100 

scale J 

In our judgment, it would he in every respect de- 
sirable for Cornell University , and any other institu- 
tion of like character, and probably also for the 
secondary schools as well, to adopt a five-division 
system of marking with the express provision that, 
in the long run, the marks given by any instructor 
must not deviate widely from the distribution just 
indicated. 

An example of a translation of a distribution of marks into a theo- 
retical curve is afforded by the work of Dr. W. S. Hall of the North- 
western University Medical College. Hall's data are quoted by subse- 
quent writers as evidence of a distribution in the binomial curve. We 
have been struck, however, with the fact that Hall's distribution is 
really a skewed curve, like our own, and that it is only by very arbi- 
trary treatment that he has succeeded in exhibiting a semblance to 
the probability curve. We believe his results confirm our own, and for 
that reason will consider them briefly at this point. 

Hall's data are derived from the marks of 2334 medical students, 
who were marked by a system of nine letters. Hall then translates 
these nine symbols into a percentile scale (see Chart 5). With this 
curve before him, he says : "That the curve derived from the rating 
of the 2334 students is really a binomial curve no fair-minded judge 
would for a moment question or doubt. We have, therefore, demon- 
strated beyond cavil that examination data is [sic] biologic data and 
obeys the laws of distribution of biologic data." 

In the next breath, however, he adds : "Certain important divergen- 
cies from strict coincidence remain yet to be explained. Why does the 
apex of the curve stand to the right of the symmetrical binomial curve, 
i. e., why is the curve of my ratings unsymmetrical? The answer is 
to be sought in two directions : 

"1. Either the examiner was too generous and habitually rated his 
students above their equitable deserts, or 

"2. The students were (in a sufiicient number of individual cases 
to influence the totals) guilty of raising their rating above what it 
should be by nature through dishonest means or extraneous aids in 



Curve as Drawn by Hauu 




5o ea 70 75 so s5 so 95 «oo 



Our "Revision of Halls- Curwe 



600- 
500— 
40d- 
300- 

200— 

100— 

— 



O 40 -45 50 55 <S0 65 70 7^ SO 85 90 95 lOO 

CiEIARTS 5 AND 6. Distribution of 2334 Marks 
Reported by Dr. W. S. Hall. 



34 



RESULTS FOR NUMEROUS COURSES 35 

quizzes, examinations and the preparation of note-books." 

As a matter of fact, Hall's curve is not a binomial curve, but a 
curve skewed to the right. This will be seen clearly enough by refer- 
ence to our Chart 6, in which we have redrawn the distribution of his 
data upon a correct scale.^ It is evident that Hall condensed the left 
end of his curve unfairly by changing the units of his abscissa at this 
point, for he has made the ranges 50-59 and 60-69 equal in extent to 
the other ranges of five points each. It is hard to understand how 
those who quote Hall with approval could have been misled by this 
feat of statistical juggling. 

Finally, the pass mark in Hall's schema is placed at 70. If this 
point were placed at 60, as is done at Cornell, his distribution would 
evidently vary still more in the direction of our own pattern curve. 



^In Charts 5 and 6, ordinates represent absolute numbers, as the 
appended scale clearly shows. 



CHAPTER IV. 

THE DISTEIBUTION OP MAEKS AT COKNELL UNIVERSITY: 
RESULTS FOR INDIVIDUAL COURSES. 

The purpose in the present chapter is to exhibit 
the distribution of marks given in 1911, course by 
course. 

The material here presented was gathered by the 
writer directly from the records of the Cornell Uni- 
versity Registrar. The plan was to begin at the 
records submitted in June, 1911, and to go back in 
each course until at least 200 separate marks had 
been secured.^ Most of the classes were large, so 
that, save for a few instances, the requisite number 
of marks was obtained within the two academic 
years 1909-1911. The charts presented in this chap- 
ter are based upon the same abscissas as in Chapter 
III, and the actual number of marks has been re- 
duced to per cents, so that direct comparison may be 
made between the different charts. The frequency 
of exemption is again based on a mark of 85, and 
the solid black area, as before, shows the distribution 
of the middle 50 per cent, of the students. The total 
number of marks is 8141 (including here the Col- 
lege of Law). The range of courses covered is as 
follows : College of Civil Engineering, one course ; 



^In five instances, however, there are fewer marks than 200 ; in one 
of these only 112 marks could be secured ; in the other four the num- 
bers lie between 190 and 200. 

37 



38 MARKING SYSTEM IN THEORY AND PRACTICE 

College of Agriculture, one course; College of 
Mechanical Engineering, three courses; College of 
Law, three courses, and College of Arts and 
Sciences, 26 courses. In the last named group the 
following subjects are included: Geology, Biology, 
English, History and Political Science, German, 
French, Mathematics, Physics, Chemistry, Botany, 
Education, Psychology and Philosophy. 

TABLE II. 

Effect of Peesonal Equation and Distributions of High 
Markers. 

Showing the Per Cent, of Students in the Several Groups. 

No. of 

Course. Marks. 0-39. 40-44. 45-49. 50-54. 55-59. 60-64. 65-69. 70-74. 

Al 263 .4 .4 2.3 4.5 1.9 13.7 11.8 16.7 

A2 257 1.2 .8 1.5 12.1 12.8 10.5 

B 208 ... 3.8 2.3 2.7 

192 .5 1.0 ... 5.2 3.7 6.3 

D 293 7 1.3 

E 216 9 2.7 

Ex- 
Course. 75-79. 80-84. 85-89. 90-94. 95-100. empt. 

Al 15.6 20.2 6.4 5.7 .4 12.5 

A2 13.6 9.8 33.9 3.8 ... 37.7 

B 10.4 13.0 23.1 26.4 18.3 67.8 

C 5.7 6.8 42.7 16.1 12.0 70.8 

D 4.4 17.4 24.9 38.6 12.7 76.2 

E 8.8 9.6 29.2 41.2 7.4 78.0 

In carrying out the general purpose of displaying 
the variability (and hence the presumptive unrelia- 
bility) of the distribution of marks in specific 
courses, we shall arrange the material in five parts. 
The first part will show the variation produced by 
the change of instructors in a given course; the 



RESULTS FOR INDIVIDUAL COURSES 39 

second will present typical distributions of ''high 
markers;'' the third, typical distributions of ''low 
markers ; ' ' the fourth will show numerous peculiari- 
ties of distribution in other curves, while the fifth 
will deal with the special problem of the marking 
system used by the College of Law. 

1. Variation produced hy change of instructors. 

A certain course in the College of Arts and 
Sciences continues throughout the year; the work 
of the first term is in charge of one professor, while 
the work of the second term is in charge of a dif- 
ferent professor. The students, with very few 
exceptions, are the same in both terms. Chart 7 
shows the distribution of marks in the first. Chart 8 
in the second term of this course, when the marks of 
several years are combined. The corresponding 
numerical data are shown in the first two distribu- 
tions (Course Al and Course A2) of Table II. The 
difference in the two distributions is at once appar- 
ent. In the first term 12.5 per cent., in the second 
term 37.7 per cent, of this class is exempt from final 
examination. In the first term 0.4 per cent of the 
students secure a mark above 95, in the second term 
no student reaches this mark. In the first term 9.5 
per cent., in the second term 3.5 per cent, of the stu- 
dents are conditioned or failed. In the first term no 
student of average ability receives a mark above 84, 
while in the second term numerous students of aver- 
age ability receive marks between 85 and 89. In fact, 
the number of students in this particular range is 
absurdly large. 

This decided difference in distribution of the ac- 
complishment of the same students in the same 
subject in the two halves of the course is evidently 



ZO- 



IC - 



CoLL&GE OF Arts <L Sciences 

rii^*5TlERM OF A Course — 

9.5ZBelowRa55 I;2.52^ Exempt 




40 45 50 55 60 65 70 75 80 85 90 95 100 

CHART 7. Distribution in Course A1. 



40 



35- 



30- 



ZS- 



Z0~ 



15 



10- 



5- 



O- 



COLLEGE or Af^T3 AND SCIENCES 

SecondTerm of Same Course 
3.5Jf Below Pass 37.7 X Exempt 




40C45 50 55 60 65 70 15 SO 65 90 35 /OO 

CHART 8. DisTBiBUTioN in Course A2, Second 
Teem of A1. 



41 



42 MARKING SYSTEM IN THEORY AND PRACTICE 

almost entirely due to the different standards of 
marking held by the two professors. 

2. Typical distributions of 'high markers,^ 

Charts 9 to 12 exhibit the distributions given by 
four typical ^high markers.' The numerical data 
for these distributions are shown in the last four 
lines of Table II. The reason for these unexpected 
deviations from the pattern distribution (Chart 4) is 
apparently different for each case. 

Chart 9 represents the distribution of a course in 
the College of Arts and Sciences in which there are 
held weekly examinations, but no final examination 
to cover the work of the entire term. It cannot be 
said positively that this fact explains the high 
marks, but it is at least a possible explanation. The 
students in the course must organize the material of 
the four lecture periods in preparation for the writ- 
ten exercise which is held in the fifth period of each 
week. But they need not organize the material of 
the entire course in preparation for a long formal 
final examination. "Whatever be the explanation, the 
result is clear enough — no student is failed or condi- 
tioned. The average student receives a mark be- 
tween 80 and 94; 67.8 per cent, of the class would 
be entitled to exemption from a final examination, 
while 18.3 per cent, of the students receive a mark 
between 95 and 100. 

Chart 10 pictures the distribution of another 
course in the same college. The professor in charge, 
who believes in passing ^ ^ handsomely, " if at all, is 
in the habit of marking the papers strictly, and 
afterward deliberately raising the marks, so as to 
throw the entire class upward. The advancing of 
the marks is done by adding a small increment to 



Z5- 



20- 



J5. 



10- 



0- 



CoLLEGE OF ArtS AND 5ciEN/CES 

Typical High Marker 

NON E Bt LOW PA55 ey. 8 ^ EXEM PT 




-fO 45 50 55 60 65 70 15 60 Q5 90 95 lOO 
CHART 9. Distribution in Course B. 



43 



-45- 



40- 



35- 



50 



Z5- 



ZO- 



15- 



/O 



5- 



CoLLEGE OF /A-RTs <1^ Sciences 

Typical High Marker 
\.5% Delow B\3S 70.© Z Exempt 




40 45 50 55 60 65 70 75 80 65 90 95 100 
CHART 10. Distribution in Coukse G. 



44 



RESULTS FOR INDIVIDUAL COURSES 45 

the original marks above 90, a larger increment to 
those between 80 and 89, a still larger one to the 
marks between 70 and 79, and so on. No marks are 
given between 55 and 59. A very small number of 
students are conditioned, and a still smaller number, 
practically negligible, are failed. A definite process 
of advancing medium-grade students to the exempt 
limit is also displayed. The result is clearly to pro- 
duce an array in which the marks of 60 to 84 occur 
with almost the same frequency, in which the marks 
/of 85 to 89 are given with disproportionate fre- 
quency, and in which altogether too many students 
are credited with an accomplishment of 90 or over. 
Exemption is granted to 70.8 per cent, of the class. 
Average students range between 80 and 94 in their 
marks. 

Chart 11 shows an array derived from a course in 
the College of Mechanical Engineering. The pri- 
mary explanation of the extremely favorable marks 
here is found in the reputation of the course as a 
^snap,' to use students' parlance. It is also possible 
that the conditions under which the work is done may 
tend less than ordinarily to check cheating on the 
part of dishonest students. Whatever factor is at 
(Work, the result is that no student is conditioned or 
failed, that all average students would be entitled 
to exemption on the 85 basis, and that 93.6 per cent, 
of the students obtain a mark of 80 or over! The 
remedy apparently is to condense the work into 
shorter compass or to incorporate the material, 
since it is so simple, in various other courses. The 
nature of the subject-matter suggests, at least, that 
it might well be included in the work given to the 
same students by another department. 



40- 



35- 



^^ College of Mechanical ENGrNEERlNG 

TVpcAL High Marker 
^5_ None Below B\ss 76. ;e% Exempt 



15-^ 



(0 — 



5- 




O 40 45 50 55 GO 65 70 75 SO ©5 90 SS lOO 



CHART 11. Distribution IN Course D. 



46 



RESULTS FOR INDIVIDUAL COURSES 47 

The most extreme case of high, marking whicli 
came within the scope of this investigation is Course 
E, shown in Chart 12. The course is one in the Col- 
lege of Arts and Sciences. The work consists of lec- 
tures, outside readings and textbook study. The 
outside reading is not specifically prescribed, but 
must simply cover a given amount of time. In many 
cases the students read books that chance to interest 
them, but which do not bear directly upon the sub- 
ject-matter of the course (though dealing, to be sure, 
with the general field of which the course forms a 
part). The lectures practically give everything 
(found in the textbook, so that the latter might really 
be dispensed with by the student without affecting 
his standing. In addition, the professor in charge 
undoubtedly overestimates the accomplishments of 
the students, or, what amounts to the same thing in 
the end, sets a low standard of performance. The 
result is that no one is ever failed or conditioned, 
that 65 is the lowest mark given, that 78 per cent. 
W'ould be exempt from the final examination, and 
that students of average ability are sure of a mark 
between 85 and 94.^ 

When we state that these four courses (B to E) 
are not selected, small-sized advanced classes, but 
large groups of undergraduates in elementary work, 
we can find no sufficient justification for the charac- 
teristics so strikingly exhibited in their graphs. 
Imagine groups of undergraduates, assembled at 
random from the student-body as they are in these 



^The tendency to mark high is inherent in human nature. Dr. Ruff- 
ner says : "A temporizing professor who loves popularity, and desires, 
like the old man in the fable, to please everybody, is sure to be guilty 
of this fault, and, like many a politician, to sacrifice permanent good 
for temporary favor." 



'^5- 



40- 



35- 



30- 



%5- 



20- 



15- 



iO- 



5- 



0- 



Colleqe of Art5 (^ Science^' 

The Highest Marker 
None "Beuow R^ss 78 ^ Exempt 




40 45 50 55 60 65 70 15 SO 85 30 95 100 
CHART 12. Distribution in Course E. 



48 



RESULTS FOR INDIVIDUAL COURSES 49 

courses, who, year after year, display such extraordi- 
nary accomplishment ! The marks charted in these 
four courses represent, collectively, 909 students, of 
which one unfortunate failed and two were con- 
ditioned. If all the courses of the university were 
patterned after these four, and if 85 were recognized, 
as it is now, as a mark of merit, then three out of 
four students would be entitled to election to the sev- 
eral honorary societies that seek for students of 
merit for enrollment in their organization. 
5. Typical distributions of 'low marhers.' 
A selected group of ^low markers' is displayed 
in Table III, Courses F to M (Charts 13 to 19), with 
the exception of Chart 14, which is placed here for 
the sake of comparison. 

Chart 13, which is perhaps the most extreme in- 
stance of low marking in our material, is a certain 
course in the College of Arts and Sciences which is 

TABLE III. 
Effect of Zeal and Distributions of Low Markees. 
Showing the Per Cent, of Students in the Several Groups. 

No. of 

Course. Marks. 0-39. 40-44. 45-49. 50-54. 55-59. 60-64. 65-69. 70-74. 

F 266 1.1 ... .7 5.3 ... 27.0 32.0 12.4 

G 353 1.7 7.3 .3 11.9 12.3 17.8 

H 226 3.1 1.8 .9 4.4 .4 26.1 16.3 13.3 

1 235 2.1 ... .9 1.7 4.2 23.4 14.$ 19.2 

J 2.34 2.1 .9 2.1 3.8 .4 21.8 16.7 19.1 

K 317 1.2 2.5 2.5 5.7 5.7 18.3 9.5 12.0 

L 208 .5 .5 1.9 3.8 1.9 18.7 10.2 16.4 

M 273 1.5 2.9 6.9 17.6 17.9 22.7 

Ex- 
Course. 75-79. 80-84. 85-89. 90-94. 95-100. empt. 

F 16.5 3.5 1.5 1.5 

G 19.8 15.0 11.9 2.0 ... 13.9 

H 12.9 9.3 7.1 2.6 1.8 11.5 

1 13.6 7.2 12.3 .9 ... 13.2 

J 8.2 6.1 14.1 4.7 ... 18.8 

K 9.8 6.6 16.7 6.6 2.9 26.2 

L 14.9 9.6 12.1 8.1 1.4 21.6 

M 13.6 12.9 2.9 1.1 ... 4.0 



35 — 



30 



College of Arts and Sciences 
Course G/ven xo Engineering Students 
7./ % Below- Pass 1.5^ Exempt 



^5• 



zo- 



15- 



/o- 



5- 



0-^ 




-fO -45 50 55 GO 65 70 75 8o 85 90 95 /OO 
CHART 13. Distribution in Course F. 



50 



20- 



15- 



10- 



CoLLEGE- OF ArT5 AND SCIENCES 

Course Given to Arts Students 
9.3^ Below Bvss f3.3^ Exempt 




O 40 45 50 55 60 65 70 75 80 85 90 95 /OO 

CHART 14. Distribution in Couese G. 
(For comparison with Chart 13, Course F, on the same 
subject.) 



51 



52 MARKING SYSTEM IN THEORY AND PRACTICE 

prescribed for students in engineering. For com- 
parison with Chart 13 we introduce here Chart 14, 
which does not happen to belong to the low-marker 
group, but which is a course on the same general 
topic, in the same department, though given by an- 
other member of the instructing staff of the depart- 
ment and primarily to arts students. This latter 
course is more theoretical; the former, that given 
to engineers, is more practical and adapted to the 
immediate problems of the man in business. It is 
generally conceded that the former course is the 
easier, but the distribution of marks shows much 
poorer accomplishment in it. This situation is in- 
teresting enough to demand a mementos attention. 
The question arises: Why is the distribution of 
Chart 13 so difPerent from that of Chart 14? The 
instructors who assigned these marks have been ac- 
customed to work together in other courses, and 
their standards, so far as can be judged, are not dis- 
similar. Again, it is impossible to argue that the 
engineering students are, as a group, inferior in 
ability to the arts students. There is left, then, the 
factor of zeal or training. Evidence furnished by the 
testimony of numerous students and corroborated 
by the opinions of the instructors themselves, makes 
it clear that the engineering students, as a group, 
look upon this prescribed course as a * necessary 
evil.' They take but a half-hearted interest in 
it, and, for the most part, strive merely to *get 
through.' The result, as Chart 13 shows, is that no 
student receives a mark above 89, that only 1.5 per 
cent, are given marks of 85 or over, that the most 
frequent mark lies between 65 and 69, and that the 
middle half of the class fall between 60 and 74; in 



45— 



40 — 


Low-Marker 


vs High-Marker 










Showing 


VARIA6IL 


iTY OF Distribution 






35- 






■ 






30- 















Z5- 






1 •' 1 






20- 












15- 






1 ' 1 1 

1 1 1 1 

1 III 

1 'II 

1 J 1 






iO- 








1 
1 
1 






5- 




r" 




1 , 


O- 


'llJ 


1 

1 , ! 


1 1 , 

1 \ ^—. , I , 



O 40 -45 50 55 eo 65 70 75 SO 35 90 95 lOO 

CHART 15. A Combination on the Same Base 
Line of Charts 12 and 13. 



53 



54 MARKING SYSTEM IN THEORY AND PRACTICE 

other words, fall, as a group, entirely below the aver- 
age of the University. 

Another point of interest: in both Chart 13 and 
Chart 14 there is displayed a decided aversion to 
giving a mark between 55 and 59 ; it is given but once 
in 619 cases. The examiner states that, in the sub- 
ject in question, it is difficult to grade a student much 
closer than 5 points ; and that it is impolitic to invite 
an argument over a mark between 55 and 59. This 
is further evidence of the impossibility of living up 
to the implications of the percentile system. 

We pause here, before passing to other instances 
of low marking, to call the reader's attention to 
Chart 15, which, because it combines upon the same 
base-line the distribution of Chart 12 (high marker) 
and Chart 13 (low marker) will serve to picture 
graphically the inequalities of the regulation mark- 
ing system in actual practice. This chart preaches 
its own sermon, so that further comment is unneces- 
sary. 

The marks displayed in Chart 16 are derived 
from a course in a department which has the general 
reputation of being the hardest marking department 
in the College of Arts and Sciences. Attention may 
be called to the mode at 60-64, to the progressive 
diminution ' of marks from 60 upwards, and to the 
fact that 10.6 per cent, of the students are condi- 
tioned or failed. A student of average ability can- 
not secure a mark above 79, while he may barely es- 
cape being conditioned. 

Charts 17 to 19 (and likewise Course L, not shown 
graphically) are all examples of relatively low 
marking, combined with an interesting tendency to 
distribute the marks so that three modes appear. 



30- 



Z'B- 



20- 



;5- 



/o. 



5 — 



College of Arts t Sciences 
Typical Low-Markek 
»o. 6^ Be LOW Pass i/.s;^ ExENiPT 




40 45 50 55 ^ 65 70 75 80 85 30 35 100 
CHART 16. Distribution in Course H. 



55 



Z5- 



ZQ. 



15- 



10 — 



5 — 



0- 



COLLEQE OF A r5.T2><£ SCIENCES 

Q.S% Below Pass \3.Z% Exempt 




O 40 45 ^0 ^5 60 65 10 75 60 S5 90 95 100 

CHART 17. Distribution in Course I. 



56 



zb- 



20- 



15- 



10- 



5- 



CouLEGE OP Arts and Sciences - 
g.3 2' Below Fas3 I8.87o E'xempt 




-K) 45 50 55 60 65 TO 75 80 85 90 95 100 

CHART 18. Distribution in Course J. 
(Note the similarity to Chart 17, another course by the 
same instructor.) 



57 



58 MARKING SYSTEM IN THEORY AND PRACTICE 

In all of them the primary mode is at the 60-64 
range; with one exception, the secondary mode is 
at the 70-74 range, and the tertiary mode at the 85- 
89 range. In other words, the commonest mark as- 
signed is that which just permits the student to pass. 
The next most common marks are those which as- 
sign the students to a rank close to the average per- 
formance of the whole student body, while the next 
most frequent mark is that which entitles the student 
to exemption. Since more students barely pass than 
reach the average mark, the curves are all skewed to 
the left. 

Charts 17 and 18 are of special interest, because 
they represent two different courses in the College 
of Arts and Sciences that are given by the same pro- 
fessor. Their similarity is striking, and is a good 
example of the influence of the personal equation in 
the distribution of marks. 

In Chart 19 the distribution of Course K repre- 
sents what may be termed a ^ staff mark. ' The course 
in question, given in the College of Arts and Sciences 
as a prescribed course to students in engineering, is 
divided into a large number of sections, and taught 
by a corps of teachers. The subject-matter of the 
course and the final examination are the same for all 
sections, but the marking of each section is in charge 
of its own teacher. All these marks are assembled 
in the one distribution. The curves show modes at 
the 60-64, 85-89 and 70-74 ranges; the average stu- 
dents' range is evidently too large (60 to 89) ; des- 
pite the mode at the 60-64 range, 26.2 per cent, of the 
class attain a mark of 85 or over. On the other 
hand, 17.6 per cent, of the students are conditioned 



20- 



»5- 



10- 



5 — 



0- 



COLLEGE OFAl?TS i^ciENCES 
17.6 Z BELOW Pas^ ze. Z X Exempt 




40 45 50 55 60 65 70 75 QO 85 90 95 /OO 
CHART 19. DiSTBiBUTioN IN Course K. 



59 



60 MARKING SYSTEM IN THEORY AND PRACTICE 

or failed. In short, the curve of distribution is too 
^flat.' 

Course L^ shows practically the same shape of 
distribution as Course K. It has the three modes 
at the same ranges as the last chart, and has 21.6 per 
cent, of the class receiving exempt marks. The aver- 
age student falls in the range from 60 to 84, and 
8.6 per cent, of the students receive either ^condition' 
or ^failure.' 

The distribution of Course M^ is less decidedly 
that of a severely marked group. The mode falls at 
70-74 rather than at 60-64, and the distribution falls 
away progressively on either side of this mode, as 
it should. However, the percentage entitled to ex- 
emption (4 per cent.) is quite small, and no member 
of the class gets 95 or over. This is a course in which 
the lectures are given by a series of different speak- 
ers, and the examination papers are marked under 
the supervision of a single man, who has general 
charge of the work of the course. 

4. Peculiarities of distrihution in other courses. 

The remaining distributions, in which the n^ark- 
ing is neither decidedly low nor decidedly high, have 
been divided into two groups, the first comprising 
unimodal, the second multimodal distributions. 

(a) The unimodal distributions. Courses N to T 
(see Table IV), have the merit of conforming, at 
least to some extent, to the theoretical distribution, 
according to which mediocrity or average accom- 
plishment is the most frequent type. In the first 
four of these courses (only one of which. Course P, 
is here shown graphically), the range of marks as- 



^Not here reproduced graphically. See Table III for details. 



Z5— 



^0— 



15— 



»0— 



5 — 



College. OF Arts >^ND cS'S'ihnceS , 
€.-*>% "Below Pass \Z.Z'/ Exempt 




O -K) 45 50 55 60 65 70 75 SO 55 90 95 100 
CHART 20. DiSTBiBUTioN in Coubse P. 



61 



62 MARKING SYSTEM IN THEORY AND PRACTICE 

TABLE IV. 

Unimodal Distributions. 

Showing the Per Cent, of Students in the Several Groups. 

No. of 

Course. Marks. 0-39. 40-44. 45-49. 

N 211 .9 .5 ... 

305 .7 .3 .7 

P 221 .4 ... .4 

Q 328 ... .3 1.2 

R 254 .8 .8 1.9 

S 254 .4 

T 112 



Course. 75-79. 

N 27.5 

18.4 

P 18.5 

Q 20.1 

R 28.8 

S 27.2 

T 11.6 



signed to students of average accomplishment is the 
same, viz., 65 to 84. In Course R the range is con- 
tracted to 65 to 79; in Course S advanced to 70-84, 
and in Course T to 80-89. In Courses R and S there 
is a slight irregularity at the 55-59 range, but the 
disturbance is too slight to disbar the distributions 
from the unimodal group. It is perhaps no accident 
that, with one exception (Course T), these unimodal 
distributions are derived from courses in pure or 
applied science. 

(h) In our second group, the multimodal curves, 
are included Courses U to DD. (See Table V). In 
details they vary considerably. We shall pass over 



50-54. 


55-59. 


60-64. 


65-69. 


70-74. 


2.4 


3.3 


8.1 


12.8 


18.5 


1.6 


1.9 


13.1 


14.8 


16.4 


1.8 


3.7 


14.5 


14.5 


20.8 


3.9 


4.6 


9.5 


17.1 


18.0 


.8 


6.0 


8.7 


17.3 


23.2 


1.5 


.8 


6.3 


9.5 


20.1 


.9 


... 




2.7 


9.8 
Ex- 


80-84. 


85-89. 


90-94. 


95-100. 


empt. 


17.5 


7.1 


1.4 


. . . 


8.5 


18.0 


9.5 


3.6 


1.0 


14.1 


13.2 


5.9 


5.9 


.4 


12.2 


12.2 


11.6 


1.5 




13.1 


7.9 


3.5 


. . . 


. . . 


3.5 


13.4 


11.0 


8.3 


1.5 


20.8 


38.4 


29.5 


4.4 


2.7 


36.6 



25 — 



ZO- 



«5 



0- f 



College of Arts and Sciences 
7.2 % Below Pass 9.9% E>^empt 




O ^O A5 50 55 eo 65 70 75 80 e>5 90 95 

CHART 21. Distribution in Course U. 



zo— 



15- 



lO — 



5- 



0- 



CoLLEGE OF Anrs and Sciences 
'0.5^ Below B\s5 18.8 /Exempt 




40 45 50 55 60 65 70 75 60 Q5 30 95 JOO 
CHART 22. Distribution in Course V. 



64 



RESULTS FOR INDIVIDUAL COURSES 65 

them rapidly, calling attention to features of interest 
in some of them. 

TABLE V. 

Multimodal Distributions. 
Shoiving the Per Cent, of Students in the Several Groups. 

No. of 
Course. Marks. 0-39. 40-44. 45-49. 50-54. 55-59. 60-64. 65-69. 70-74. 

U 262 A A 1.1 3.0 2.3 16.5 13.4 22.8 

V 191 2.1 1.1 3.7 .5 3.1 16.7 9.5 14.3 

W 207 .5 ... 1.4 2.9 .5 15.9 16.5 18.4 

X 225 1.8 4.9 17.8 11.1 10.2 

Y 251 ... .8 .4 5.2 3.2 14.7 13.7 18.6 

Z 195 ... .5 .5 ... .5 1.5 1.5 9.8 

AA 232 1.3 ... 8.3 10.3 20.6 

BB 215 1.4 5.5 2.8 15.7 19.6 16.3 

CC 228 .4 .4 1.8 .4 2.2 3.1 7.9 18.0 

DD 198 ... 1.5 1.0 4.1 .5 10.6 13.2 11.6 

Ex- 
Course. 75-79. 80-84. 85-89. 90-94. 95-100. empt. 

U 18.4 11.8 9.1 .8 ... 9.9 

V 19.8 10.4 13.1 4.2 1.5 18.8 

W 17.4 4.3 19.3 2.9 ... 22.2 

X 15.1 7.1 18.6 11.6 1.8 32.0 

Y 12.7 9.6 14.3 6.4 .4 21.1 

Z 30.3 22.1 30.8 2.0 .5 33.3 

AA 12.9 28.1 14.6 3.9 ... 18.5 

BB 18.7 11.2 7.0 1.0 ... 8.8 

CO 17.1 16.3 19.2 11.0 2.2 32.4 

DD 15.1 13.2 17.2 7.5 4.5 29.2 

Chart 21 is the array for a course in the College 
of Arts and Sciences, whose professor is generally 
reputed to be a fair marker. His curve of distribu- 
tion varies, however, from the pattern we have rec- 
ommended ; first, in that it shows a general tendency 
to fall short in the frequencies in the upper end of 
the scale (no mark above 94, only 9.9 per cent, 
exempt, and no average student gaining a mark 
above 79) ; second, in that it shows a disproportion- 



Ze- 



is— 



/o— 



5— - 



CoLnEGE OF Arts and Sciences 
5.3;^Delow Pass Z7..Z% Exempt 




O -»0 45 50 55 <SO 65 70 75 80 85 90 95 /OO 

CHART 23. Distribution in Course W. 



66 



RESULTS FOR INDIVIDUAL COURSES 67 

ate tendency to give marks between 60 and 64. Some 
of the marks in this range should have fallen in the 
55-59 range, and some probably in the 65-69 range. 

Chart 22 resembles Chart 19 in being a composite 
of marks for several sections of the same course, the 
present distribution being that of a language course 
in the College of Arts and Sciences, which is run in 
four sections. The two distributions have a certain 
amount of similarity, notably in the low frequencies 
assigned to the middle range of accomplishment. 
Medium-grade students range in marks from 60 to 
84. The present curve has three modes, at 60-64, 
75-79 and 85-89, respectively. 

The peculiarity of Chart 23 (Course W), a science 
course, is in the curious ^hole' at the 80-84 section of 
the scale. To counterbalance this failing there ap- 
pears a second curiosity — the range 85-89 forms the 
primary mode of the distribution. Again, despite 
the fact that 22.2 per cent, reach the grade of 85 or 
over, no one exceeds 94. Finally, the frequency of 
the marks between 60 and 79 is virtually constant for 
each section of ^ve points. 

Another language course is shown graphically in 
Chart 24 (Course X), which displays considerable 
irregularity and deviation from the distribution to 
be expected. Exemption is rather freely accorded 
(32 per cent.), and average students are spread over 
a range from 65 to 89. This failure to perceive the 
homogeneous character of this group of medium 
worth is the fundamental defect of the curve. A 
fondness for 60-64 is another evident characteristic, 
due, evidently, to pushing over the 60 mark those 
who should be conditioned, only 15 out of 225 are 
conditioned, while no one fails. 



zo- 



15- 



»0 



5 — 



College of At?ts and Sciences 

6.7 % DelOWFASS 3Z% ExEMF^r 




40 45 50 55 60 65 70 75 eO 65 90 95 iOO 
CHART 24. Distribution in Course X. 



68 



30 



Z5- 



20- 



15 



10 



College of Mechanical Engineerino 
1.5 5!1 Below Pa§5^ 18.5^ Exempt ' 



0— r- 



1=1 




40 45 50 55 60 65 70 75 80 85 90 95" /oo 
CHART 25. DiSTBiBUTioN in Course AA. 



70 MARKING SYSTEM IN THEORY AND PRACTICE 

A somewhat similar criticism could be passed 
upon Course Y (not reproduced graphically), a 
course in the College of Arts and Sciences, in which 
too many 80-to-84-grade students are pushed for- 
ward to exemption, and in which the 60-64 range is 
also too frequently used. 

A member of the faculty who nearly deserves to 
be grouped in our second division, the high-markers, 
is responsible for Course Z (not reproduced, save in 
Table V). Exemption is gained by one-third of the 
class ; only 1.5 per cent, of the students are failed or 
conditioned, while the average student never falls 
below 75, but may, indeed, win a mark of 89. 

The course in the College of Mechanical Engineer- 
ing (Course AA), displayed in Chart 25, is not one 
that should cause worry on the part of the student. 
"While no one gets above 94, yet no one fails, and but 
1.3 per cent, of the students are conditioned. The 
average student ranks between 70 and 84. Here the 
tendency is apparently to avoid the range 75-79 in 
favor of the range 80-84. 

Course BB (see Table V) is from one of the large 
introductory courses in pure science. In general, 
the distribution inclines towards the lower marks, 
so that only 8.8 per cent, would reach exemption ; no 
one exceeds 94, and 65-69 is the mode. 

Another large elementary course in science is dis- 
played in Chart 26 (Course CC). Save for a too 
high number of those exempted (32.4 per cent.), the 
distribution is one of the best found in our data. 

In Chart 27, from the College of Arts and 
Sciences, the examiner has avoided the 55-59 range. 
The proportion exempted (29.2 per cent.) is too 
high, but the general form of the distribution is 



XO- 



IS — 



lO- 



CoLlETGe of Arts and Sciences 
5.^ J' Delovv "Rve-s 3;2.4^ Exemft 




O -fO -fS 50 55 eO 65 70 75 80 85 90 95 foo 

CHART 26. Distribution in Coubse CC. 



71 



72 MARKING SYSTEM IN THEORY AND PRACTICE 

otherwise not bad. TMs chart may be profitably 
compared with Chart 12 (p. 48), as the two come 
from the same department, apply to the same stu- 
dents, but are given by two different professors. The 
difference in the standards of accomplishment held 
by these two members of the faculty yields further 
evidence, if such be needed, of the inequalities that 
prevail in the marking system at present in force at 
Cornell. 
5. The marMng system of the College of Laiv, 
In the College of Law there prevails a system of 
marking that is radically different from the percen- 
tile system that we have just been discussing at 
length. The Law School system embraces six dis- 
tinct marks, but, unfortunately, these six marks do 
not conform, either in intention or in practice, with 
the restricted-unit systems that were discussed in 
Chapter II. Nor does it appear that any effort has 
been made by those who introduced this system to 
relate it definitely to the percentile system which it 
replaced, and which still prevails elsewhere in the 
University. The translation into the regular Uni- 
versity percentile system of the six symbols — EE, 
E, G, Pj P-60 and Cond. — ^which are in use in the Col- 
lege of Law, has been arranged in the charts which 
follow, in accordance with the statements of their 
values, as furnished by the Dean of that College.^ 

Law School Marks Cond. P-60 P G E EE 

Approximate Equivalents. . .Below 60 58-63 60-74 75-89 90-98 99-100 



^It appears that a mark (in the numerical system) of 60 might be 
either P-60 or P in the College of Law. It is explained that the 
papers are marked in numerical terms and then translated into the 
six symbols. If a final paper warranted 62, the instructor would 
report the paper as P if the class work was acceptable, but as P-60 
if both class work and final examination were inferior. 



Zo^ 



College op" Arts and ScrENCES 
7. ' % Delow B^S£> Z^. z Z Exempt 



15 



5- 



o — 




O 40 -*5 60 55 60 G5 7o . 75 80 85 SO 35 lOO 

CHART 27. DiSTEiBUTioN in Couese DD. 
(Compare with Chart 12 by another instructor in the 
same department.) 



73 



74 MARKING SYSTEM IN THEORY AND PRACTICE 

The data from the College of Law, Table VI and 
Courses EE, FF and GGr, are derived from one first- 
year and two second-year courses. In examining 

TABLE VI. 

Distribution of Marks in Three Courses of the College of Law. 

Showing the Per Cent, of Students in the Six Groups. 

No. of Ex- 
Course. Marks. Cond. P-60. P. G. E. EE. empt. 

EE 251 20.3 17.2 33.6 21.8 7.1 ... 7.1 

FF 238 19.3 131 37.0 16.4 13.0 1.2 14.2 

GG 222 12.1 16.3 36.5 27.9 6.3 .9 7.2 

these charts the reader must remember that height 
of column, not area, is significant. For instance, the 
large areas on the left, which represent simply 
^' below 60,'' are here plotted on the same abscissas 
used in constructing the graphs for other courses in 
the University. On the different charts P-60 is rep- 
resented by the range 60-64, while EE is represented 
by the range 95-100 (not strictly according to the 
numerical equivalence just quoted). Since no divi- 
sion is made at 85, it is impossible to compute the 
frequency of exemption on the same basis as for the 
other colleges. This frequency has been calculated, 
however, as if it included the marks E and EE; as if, 
in other words, it included marks of 90 and over. 
Naturally, this frequency is small — 7.1 to 14.2 per 
cent. 

We may begin with the consideration of Chart 
28, the first-year course. The chief feature is the 
large number of students conditioned (20.3 per 
cent.), and the small number reaching the two upper 
grades (7.1 per cent, get E, none gets EE), What is 
the explanation of this extraordinary situation! 
From the evidence at our command it appears that 
the professor in charge is a severe marker, who be- 



College op Law 
Z0.3 % 5EL0W Fass 7.1% Exempt 



35— 



30- 



%5-- 



20- 



IS- 



JO — 



5 — 



0- 



O 40 45 5P 55 60 65 70 75 50 35 30 95 lOO 
CHART 28. DiSTBIBUTION IN COUBSE EE. 



75 



76 MARKING SYSTEM IN THEORY AND PRACTICE 

lieves in conditioning regularly a certain percentage 
of his class ; that the examination is searching ; and, 
finally, that the nature of the work is different from 
what the beginner has encountered elsewhere in his 
career, whether in high school or college. The ques- 
tion may at least be raised whether some change 
ought not to be made in the conditions under which 
this course is given, so that not so many as one man 
in five would fail. 

In Course FF (not shown graphically), a second- 
year law course, the conditions are practically iden- 
tical, so far as conditioned students are concerned, 
and the curves elsewhere are closely similar, save 
that in this course the frequency of the two higher 
marks, E and EE, is somewhat increased at the ex- 
pense of the mark (r. 

In our last graph. Chart 29, we show the distribu- 
tion of another second-year law course (Course 
GG), whose examiner is reputed to be the highest 
marker in the College of Law. About 12.1 per cent, 
of the class is conditioned, but P remains the most 
frequent mark. 

The trouble with these law curves is evident 
enough. They use a better number of symbols than 
other colleges in the University, but these symbols 
are, in our opinion, improperly rated. The most 
frequent mark is "Pass,'' which means inferior to 
the average, as we have seen in our discussion of the 
theoretical considerations underlying the use of a 
limited-division marking system. The mark G 
(good) should be changed to M (medium), or some 
other symbol of mediocrity, leaving two divisions 
above for superior and excellent students, and this 
mark M should be more frequent than P. 



-K5 — 



55- 



30- 



25- 



ZO- 



15-— 



5- 



CoLLEGE or Law 
iz.i % Below T^ss 7.Z% Exempt 




O -K) 45 50 55 GO 65 70 75 60 85 30 95 100 

CHART 29. DiSTBiBUTiON in Coubse GG. 



77 



CHAPTEE V. 

SUMMARY AND CONCLUSIONS. 

Our investigation has led us to the following con- 
clusions — some of them confessedly theoretical and 
deductive, others incontestable inductions from 
carefully compiled data. 

1. The marking system of any institution of 
learning plays so important a role in the activities 
of the institution that its machinery, its significance 
and particularly its reliability is a matter that de- 
serves and demands patient and impartial study. 

2. Marks may be based upon performance, upon 
^ability, or upon accomplishment. The last named is, 

save under unusual circumstances, the quality on 
which the marks should be based. 

3. It is highly probable that ability, whether in 
high school or in college, is distributed in the form 
of the probability curve. It is at least possible, and 
we think it very probable that accomplishment, how- 
ever, is distributed, under conditions commonly pre- 
vailing in school and college, in the form of a curve 
skewed toward the upper range. 

4. The number of symbols proposed for record- 
ing degrees of accomplishment ranges from two to 
one hundred. Every theoretical consideration and 
many practical considerations favor a five-division 
system, based in essence upon five qualities of ac- 

79 



80 MARKING SYSTEM IN THEORY AND PRACTICE 

complishment, viz., excellent, superior, medium, in- 
ferior and very poor (failure). 

5. A curve compounded from more than 20,000 
marks shows that at Cornell University the * pat- 
tern' distribution is that of a curve skewed toward 
the upper range, with a mode at 75-79, and the aver- 
age at approximately 75 (60 being the pass-mark). 
The frequency of deviations above and below the 
mode decreases regularly on either side, save for a 
disturbance at the 60-point. This disturbance is 
caused partially by an effort on the part of some 
students to do just enough work to pass, but still 
more by a strong tendency of examiners to advance 
marks lying between 55 and 59 to 60 or over. 

6. The data obtained for 31 individual courses 
(7430 marks) shows that the marks of members of 
the instructing staff are strongly affected by a per- 
sonal equation — so much that typical distributions 
taken from high markers and from low markers 
show no similarity whatsoever. 

a. The percentage of students obtaining 85 or 
over (a range which, in many classes, entitles the 
student to exemption from final examination, and 
which, by assumption, indicates a quality of work 
superior to that of the medium student) falls to 1.5 
per cent, in one class, and rises to 78 per cent, in 
another class in the University. 

h. Students of medium accomplishment (who by 
definition are relatively like one another ip merit) 
are by some examiners rated between 85 and 94, but 
by other examiners 60 to 74. Again, these students 
are by some instructors spread over a range of 30 
points, by others limited to a range of 10 points. 

c. The marks of the same students, continuing the 



SUMMARY AND CONCLUSIONS 81 

same subject, sliow a different form of distribution 
when the instructor is changed. 

d. Distributions which show radical divergencies 
in form and tendency may be obtained from the 
records of two teachers engaged in precisely the 
same work. 

7. These and other variations in the assignment 
of marks need not always be laid at the door of the 
instructor. We have shown how the same subject, 
taught to different groups of students, e. g., to arts 
students and to engineering students, may yield a 
differently formed curve of distribution. 

8. The curves for individual courses are often 
multimodal. In other words, there are two or more 
ranges in the marks which occur with a frequency 
greater than that of the ranges on either side of 
them. Commonly, these modes are located at three 
points, viz., 60-64, 75-79 and 85-89. The first of these 
is due to the tendency indicated above (Conclusion 
5) : the second is the normal 'peak' of average ac- 
complishment; the third is due to a tendency, an- 
alogous to the first, to increase the number of stu- 
dents who are exempt from final examination, %, e., 
to advance marks from 80-84 to 85 or over. 

9. There appears to be a tendency for marks in 
courses in pure science and applied science to con- 
form more. closely to the theoretically presimiptive 
distribution than do marks in other courses. But 
this generalization is insecure because, after all, we 
have charted in detail only 34 out of the several hun- 
dred courses offered in the University. 

10. The marking system employed in the College 
of Law has the merit of using a restricted number 
of symbols, but it does not conform to the theoretical 



82 MARKING SYSTEM IN THEORY AND PRACTICE 

curve of distribution, nor was it designed with the 
proper theoretical considerations (discussed in 
Chapter II). 

11. The marking system used by most faculty 
members for recording the work of graduate stu- 
dents (two divisions, satisfactory and not satisfac- 
tory,) is not to be recommended for use with under- 
graduates, at least under the conditions that now 
prevail. 

12. We recommend that every institution of 
learning, at least every high school and college, 
adopt a five-division marking system, based upon a 
distribution ivMch should, in the long run, not de- 
viate appreciably from the following: Excellent, 
3 per cent.; superior, 21 per cent.; medium, 45 per 
cent.; inferior, 19 per cent; very poor, 12 per cent. 
For purposes of administration the very poor group 
may be subdivided so that approximately 11 per cent, 
shall be conditioned, and 1 per cent, shall fail. This 
distribution conforms well with theoretical require- 
ments, and coincides closely with the present prac- 
tice of Cornell University, as shown by the tabula- 
tion of 20,348 marks, drawn from a period of three 
different years and from 163 courses. It is impor- 
tant to note that, by this proposed system of mark- 
ing, the meaning of each mark is exactly defined, 
and in the only satisfactory way by which a mark 
can be defined, viz., in terms of the frequency with 
which it can be secured by students under actual 
working conditions. 

13. Furthermore, as Meyer (9, p. 664) advocates, 
in order to ensure the working of the system the 



SUMMARY AND CONCLUSIONS ' 83 

distribution actually given should be tabulated at 
stated intervals, say biennially, and the distribution 
should be made public, so that every examiner shall 
know to what extent he conforms to the principles 
on which the system is based. 



BIBLIOGRAPHY. 

(1) Cattetx, J. M. Examinations, Grades and Credits. Popular 

Science Monthly, 66 : 1905. p. 367. ^>^ 
""(2) COLVIN, S. S. Marks and the Marking System as an Iricentive 

to Study. Education, 32 : 1912. May, p. 560. . t^^ 
\d) Dearborn, W. F. School and University Grades, Bulletin of 

University of Wisconsin. 1910, No. 368. 
(4) Foster, W. T. Administration of the College Curriculum. 

Boston : H. Mifflin Co. 1911. Chap. 13. 
jv (5) Hall, W. S. A Guide to the Equitable Grading of Students. 

School Sciemce and Math., 6 : June, 1906. 

(6) HuEY, E. B. Retardation and the Mental Examination of Re- 

tarded Children. Journal of Psycho- Asthenics, 15 : Sept. 
and Dec, 1910, p. 31. 

(7) JUDD, C. H. On the Comparison of Grading Systems in High 

Schools and Colleges. School Review,!^: 1910, p. 460. 
--^(8) Meyer, M. The Grading df Students. /8'cn'ewce, ]V. S., 28: 

1908, p. 243. 
(9) Meyer, M. Experiences with the Grading System of the Uni- 
versity of Missouri. Science, N. S., 33 : 1911, p. 661. 
j^-K'iO) Sargent, E. B. Education of Examiners. Nature, 70: 1904, 

p. 63. 
%(11) Smith, A. G. A Rational College Marking System. Journal 

of Educational Psychology, 2 : 1911, p. 383. 
•^'(12) Steele, A. G. Training Treachers to Grade. Pedagogical 

Seminary, 18 : 1911, p. 523. 
^13) Stevens, W. L. American Titles and Distinctions. Popular 

Science Monthly, 63 : 1903, p. 310. 



85 



INDEX. 

Accomplishment, how determined, 10; distribution 

of, 14 fe., 79. 
Arts and Sciences, College of, 42 ff., 47, 49 ff., 65 ff. 
Cattell, J. McK., 5, 14, 19, 27, 30, 31, 32. 
Charts, general explanation of, 23 ff. 
Colvin, S. S., 6. 
Conclusions, 79 ff. 
Cornell University, marking system in, 15 f., 23; 

distribution of marks 1902-03, 21 ff.; 1903-04, 

21 ff.; 1910-11, 22 ff., 37 ff. 
Dearborn, W. F., 5, 14, 19, 27, 31, 32. 
Distribution of marks, theoretical, 11 ff., 79 ff. ; ideal, 

33, 82 ; unimodal, 60 ff . ; multimodal, 62 ff ., 81. 

See high-markers ; low-markers. 
Elimination, effect on distribution, 12 ff. 
Foster, W. T., 5. 
Hall, W. S., 5, 33 ff. 
High-markers, 42 ff. 
Judd, C. H., 13. 
Law, College of, 72 ff., 81 f. 
Low-markers, 49 ff. 
Marking system, theory of, 9 ff . ; two division, 16 f ., 

82; three division, 17 f. ; four division, 18; ^ve 

division, 18 f., 30 ff., 79 f., 82; Percentile, 6, 19 f. 
Mechanical Engineering, College of, 45, 69 f . 
Missouri, University of, 14, 20. 
Meyer, M., 5, 6, 7, 13, 14, 19, 20, 27, 30, 31, 32, 82. 

87 



88 Index. 

Native ability, distribution of, 11 fP. 

Northwestern University Medical College, distribu- 
tion in, 33 fe. 

Performance, marks based on, 9 f ., 79. 

Probability curve, 11. 

Euffner, 47. 

Science, courses in, 62, 70, 81. 

Skewed curve, 12 f . ; interpretation of, 31 ff. 

Smith, A. G., 5, 10 f ., 14. 

Starch, D., 20. 

Steele, A. G., 27. 

Tables, I, 22 ; II, 38 ; III, 49 ; IV, 62 ; V, 65 ; VI, 74. 

Units, best number of, 16 if., 79 f . 

Variations produced by change of instructors, 39 ff., 
80 f. 



SEP 17 1913